Input-to-state stability of discrete-time memristive neural networks with two delay components☆
Introduction
Memristor was originally postulated by Leon Chua [1] and realized by Hewlett-Packard researchers [2]. As the fourth basic element of electrical circuits, memristor not only likes neurons synapses in the human brain [3], but also has a lot of other characteristics, such as low power, high density and good scalability [4]. Prezioso et al. have experimentally demonstrated an artificial neural network using memristors integrated into a dense, transistor-free crossbar circuit [5]. S.G. Hu and Y. Liu et al. have successfully constructed a Hopfield network using HfO2 memristors and peripheral devices to realize the associative memory that is capable of retrieving a piece of data upon presentation of partial information from that piece of data [6]. Compared with conventional recurrent neural networks, memristive neural networks (MNNs) possess many merits in various fields, such as secure communications, associative memory, information processing and so on [7], [8], [9]. Therefore, most of the researchers focus on modeling neuron networks with memristors [10], [11], [12], [13].
It is well known that the time delays are frequently a source of instability, and the time delays in neurons are usually time variant due to the finite switching speed of amplifiers in the electrical circuit [14], [15]. Recently, many papers about delay systems have been reported [16], [17], [18]. Signals transmitted from one point to another may experience a few segments of networks, which can induce successive delays with different properties due to variable network transmission conditions [19]. Thus, the dynamical behaviors of MNNs with multiple delay components become one of the most important problems. However, there are only a few literatures about the dynamical behaviors of MNNs with two delay components [20], [21], [22], [23]. The quadratical stability and extended dissipative conditions for the MNNs with two additive time-varying delays have been proposed in [20]. The synchronization of MNNs with two delay components based on second order reciprocally convex approach has been investigated in [23]. The dynamic delay interval (DDI) method is first proposed to deal with stability problem of continuous-time neural networks in [19], which makes it to get less conservative criteria of the systems with two time-varying delay components dependent or independent on lower bounds of derivatives of time delays.
Recently, input-to-state stability (ISS) for MNNs has become a hot research topic [24], [25], which is more general than the traditional stability since the ISS properties imply not only that the unperturbed system is asymptotically stable in the Lyapunov sense but also that its behavior remains bounded when its inputs are bounded [26], [27]. Sufficient conditions for the existence of an ISS Lyapunov function have been obtained as the interconnection of many subsystems in [28]. Mean square exponential ISS stability of stochastic memristive complex valued neural networks is investigated in [29]. Some novel algebraic criteria for ensuring the ISS stability of memristor-based complex-valued neural networks are derived by employing the differential inclusion theory and nonsmooth analysis in [30]. Considering the MNNs that take into account the stochastic effects and time-varying delay, the authors in [31] establish sufficient conditions for both mean-square exponential ISS stability and mean-square exponential stability. In todays digital world, the discrete-time neural networks possess a better position than their continuous-time analogs, so the investigation of discrete-time neural networks has been paid as the same attention as that of continuous-time neural networks [32]. However, despite the clear engineering insights, the discrete-time MNNs have gained only a little attention [33], [34], [35], [36], mainly due to the mathematical difficulties in quantifying and tackling the state-dependent switching behaviors in the discrete-time setting [37]. To the best of our knowledge, few researches focus on the input-to-state stability of DMNNs.
This paper concentrates on the ISS stability problem of DMNNs with two delay components. The main contributions of this paper are summarized as follows:
- 1.
A tractable model of DMNNs with two time-varying delay components is presented in order to reflect the engineering practice. It should be noted that the ISS stability problem of DMNNs with single time delay can be regarded as a special case of this paper.
- 2.
Two time-varying delay components are taken into account in the framework of DMNNs for the first time. The DDI method is introduced to handle the delay-dependent stability of DMNNs with two time-varying delay components, which makes it possible to find optimal dynamic upper and lower bounds of a time varying delay, and brings the conservatism as little as possible to the stability criterion for the DMNNs.
- 3.
Refined Jensen-based inequalities are used to estimate double summation terms, these inequalities are helpful to find a more accurate upper bound for the difference of Lyapunov–Krasovskii functionals (LKFs).
The remaining part of this paper is organized as follows. In Section 2, the model of DMNNs is formulated by applying semidiscretization technique, and some necessary preliminaries are presented. The main theoretical results are derived in Section 3. In Section 4, numerical simulations are included to verify the effectiveness of the derived theoretical results. Section 5 draws the conclusion.
Notation: The notations used throughout this paper are standard. and respectively, denote the set of non-negative real numbers, n-dimensional Euclidean space, m × n real matrices, integers and all non-negative integers. For a < b, stands for the set of integers between a and b. ‖ · ‖ refers to the Euclidean vector norm for a given matrix or vector. The symbol λmax (P) and λmin (P) denote the maximum and minimum eigenvalues of matrix P, respectively. diag{⋅⋅⋅} is the block diagonal matrix. The notation P > 0(P ≥ 0) means that P is the real symmetric matrix and positive definite (semidefinite). and XT denote the inverse and transpose of matrix X, respectively. and are the identity matrix and zero matrix with appropriate dimensions, respectively. The notation ⋆ is the symmetric block in one symmetric matrix. Recall that a function is said to be of class if it is continuous, strictly increasing and satisfies and we write it as . A function ω is of class if and also satisfies ω(t) → ∞ as t → ∞, and we write it as . A function is said to be of class if γ(s, t) is of class on the first argument s and is decreasing on the second argument t, called . ⌊X⌋ and ⌈X⌉ round the elements of X to the nearest integers towards minus and plus infinity, respectively.
Section snippets
Problem description and preliminaries
In this paper, we consider the following class of continuous-time memristive neural networks with two additive time-varying components:for all t ≥ 0, where denotes the state variable of neurons; is the external input vector; and are two additive time varying delays; and
Main results
In this section, by using the LKFs method and linear matrix inequality (LMI) technique, we will derive the ISS criterion for the DMNNs (3) in the following theorem. In what follows, for all we denote the summation term as: Theorem 1 Assume that Assumption 1 holds. For given constant integers h1m ≥ 0, h2m ≥ 0, h1M ≥ 1, h2M ≥ 1, d1m < 0, d2m < 0, d1M > 0, d2M > 0 and constants α, β satisfying (7), DMNNs (3) is ISS if there exist
Numerical examples
In this section, we give two examples to demonstrate the effectiveness of the proposed methods. Example 1 Consider a two-neuron DMNNs (3) with and memristive synaptic weight are as follows:
Conclusion
In this paper, a class of discrete-time memristive neural networks with two additive delay components has been investigated. By constructing suitable LKFs and using dynamic delay interval method, Refined Jensen-based inequalities and free-weighting matrices, we have obtained some novel sufficient conditions to ensure the input-to-state stability for the addressed system. Moreover, two numerical examples and their simulations are given to show the effectiveness of the theoretical results. A key
Qianhua Fu Qianhua Fu received the B.S. degree in electronic information engineering from Chongqing University of Technology, China, in 2003, and received the M.S. degree in communication and information systems from University of Electronic Science and Technology of China (UESTC) in 2010. He was a R&D engineer in HUAWEI company from 2010 to 2014. He is currently pursuing the Ph.D. degree in information and communication Engineering, UESTC and working as an engineer at Xihua University. His
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Qianhua Fu Qianhua Fu received the B.S. degree in electronic information engineering from Chongqing University of Technology, China, in 2003, and received the M.S. degree in communication and information systems from University of Electronic Science and Technology of China (UESTC) in 2010. He was a R&D engineer in HUAWEI company from 2010 to 2014. He is currently pursuing the Ph.D. degree in information and communication Engineering, UESTC and working as an engineer at Xihua University. His main research interests are memristor neural network, RF circuits and systems for wireless communications, and signal processing in modern communication.
Jingye Cai Jingye Cai received the B.S. degree from Sichuan University in 1983, and the M.S. degree from the University of Electronic Science and Technology of China (UESTC), Chengdu, in 1990. He is currently a professor with the School of Software and Information Engineering, UESTC. His research interests include nonlinear circuits and systems (memristor), communication signal processing, frequency synthesis, RF and wireless systems.
Shouming Zhong Shouming Zhong was born on November 5, 1955. He graduated from University of Electronic Science and Technology of China, majoring Applied Mathematics on Differential Equation. He is a professor of School of Mathematical Sciences, University of Electronic Science and Technology of China, since June 1997-present. He is the director of Chinese Mathematical Biology Society, the chair of Biomathematics in Sichuan, and editor of Journal of Biomathematics. He has reviewed for many journals, such as Journal of Theory and Application on Control, Journal of Automation, Journal of Electronics, and Journal of Electronics Science. His research interest is stability theorem and its application research of the differential system, the robustness control, neural network and biomathematics.
Yongbin Yu Yongbin Yu received his M.S. degree and Ph.D. degree in circuits and systems from University of Electronic Science and Technology of China (UESTC) in 2004 and 2008. He is currently an associate professor with the School of Software and Information Engineering, UESTC. His current research interest covers nonlinear circuits and systems (memristor), artificial intelligence (neural networks, genetic algorithm), VLSI physical design, modern control theory and its application.
Yaonan Shan Yaonan Shan was born in Anyang, China. She received B.S. degree from Zhongyuan University of Technology, Zhengzhou, China, in 2015. Now she is working towards the Ph.D. degree in School of Information and Software Engineering at the University of Electronic Science and Technol- 480 ogy of China. Her current research interests including stability theorem, the robustness stability, time-delay system and neural networks.
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This work was supported in part by the National Natural Science Foundation of China under Grant 61533006, in part by the Scientific Research Project of Sichuan Provincial Education Department (18ZB0572), in part by the Research Fund for International Young Scientists of National Natural Science Foundation of China under Grant 61550110248.